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c 2010
THANG TIEN NGUYEN
ALL RIGHTS RESERVED
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ABSTRACT OF THE DISSERTATION
Sliding Mode Control for Systems with Slow and Fast Modes
by THANG TIEN NGUYEN
Dissertation Director:
Professor Zoran Gajic
This dissertation addresses the problems of sliding mode control for systems with slow
and fast dynamics. Sliding mode control is a type of variable structure control, where
sliding surfaces or manifolds are designed such that system trajectories exhibit de-
sirable properties when confined to these manifolds. A system using a sliding mode
control strategy can display a robust performance against parametric and exogenous
disturbances under the matching condition (Drazenovics condition). This property
is of extreme importance in practice where most systems are affected by parametric
uncertainties and external disturbances.
First, we investigate a high gain output feedback sliding mode control problem for
sampled-data systems with an unknown external disturbance. It is well-known that
under high gain output feedback, a regular system can be brought into a singularly
perturbed form with slow and fast dynamics. An output feedback based sliding surface
is designed using some standard techniques for continuous-time systems. Next, we con-
struct a discrete-time output feedback sliding mode control law for the sliding surface.
The main challenge in this work is the appearance of the external disturbance in the
control law. A remedy is to approximate the disturbance by system information of the
previous time sampling period. The synthesized control law is able to provide promising
results with high robustness against the external disturbance, which is demonstrated
by the bounds of the sliding mode and state variables. These characteristics are further
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improved by a method which takes into account system information of two previous
time instants in order to better approximate the disturbance. The stability and ro-
bustness of the closed-loop system under the proposed control laws are analyzed by
studying a transformed singularly perturbed discrete-time system.
The second topic of the thesis is to study sliding mode control for singularly per-
turbed systems which exhibit slow and fast dynamics. A state feedback control law
is designed for either slow or fast modes. Then, the system under that state feedback
control law is put into a triangular form. In the new coordinates, a sliding surface
is constructed for the remaining modes using Utkin and Youngs method. The sliding
mode control law is synthesized by a control method which is an improved version of the
unit control method by Utkin. Lastly, the proposed composite control law consisting
of the state feedback control law and sliding mode control is realized. It is shown that
stability and disturbance rejection are achieved. Our results show much improvement
when compared to the other works available in the literature on the same problem.
The problem of sliding mode control for singularly perturbed systems is also ad-
dressed by the Lyapunov approaches. First, a state feedback composite control is
designed to stabilize the system. Then, Lyapunov functions based on the state feed-
back control law and the system dynamics are employed in an effort to synthesize a
sliding surface. Two sliding surfaces and two sliding mode controllers are proposed in
this direction. Theoretical and simulation results show the effectiveness of the proposed
methods. Like composite approaches, the Lyapunov ones provide asymptotic stability
and disturbance rejection.
We also study singularly perturbed discrete-time systems with parametric uncer-
tainty. Proceeding along the same lines as in the continuous-time case, we propose two
approaches to construct a composite control law: a state feedback controller to stabi-
lize either slow or fast modes and a sliding mode controller designed for the remaining
modes. It is shown that the closed-loop system under the proposed control laws is
asymptotically stable provided the perturbation parameter is small enough.
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Acknowledgements
The writing of the dissertation could not have been complete without the enormous
support of numerous people. I would like to acknowledge all those people who have
made contribution to the completion of this dissertation.
First, my great thanks go to my advisor, Prof. Zoran Gajic who has spared no time
and energy to teach and advise me for years. His profound ideas and guidance have
helped me to learn how to do research and write scientific papers. I will cherish all of
experiences that I gain when working with him. In addition, I deeply acknowledge Prof.
Gajics wife, Dr. Verica Radisavljevic for numerous discussions and lectures related to
my study.
I also would like to extend my warm gratitude to Prof. Wu-Chung Su from National
Chung-Hsing University, Taiwan, my doctoral dissertation co-advisor. His continuous
guidance and encouragement lead me to learn sliding mode control and face with in-
teresting problems which play a pivotal role in my dissertation.
I am greatly grateful Prof. Dario Pompili, Prof. Predrag Spasojevic who serve as
committee members for their insightful comments and constructive criticism that help
finish the dissertation.
I am also indebted to many people on the faculty and staff of the Department of
Electrical and Computer Engineering. Particularly, I would like to extend my thanks
to Prof. Yicheng Lu and Ms. Lynn Ruggiero for their support and practical advice to
pursue my studies in the department. I am especially thankful to Prof. Narinda Puri,
Prof. Sophocles Osfanidis, Prof. Eduardo Sontag, Prof. Roy Yates for inspiring me to
learn and explore new areas.
Several friends have been with me through these challenging but rewarding years.
Their concern and support have helped me overcome obstacles and stay focused on my
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direction. I greatly appreciate their friendship and assistance. Bo-Hwan Jung, Gun-
Hyung Park, Prashanth Gopala, Maja Skataric, Meng-Bi Cheng, Phong Huynh, Tuan
Nguyen, and many others.
Most importantly, my dissertation would not have been possible without the en-
couragement and support from my family. I would like to dedicate my dissertation to
my parents and my siblings who are a source of care, love, support and strength during
my graduate study.
Finally, I deeply appreciate financial support from the Vietnam Education Founda-
tion and the Department of Electrical Engineering that have funded my research and
study at Rutgers University.
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Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Singularly Perturbed Systems . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3. Literature on Relevant Works . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4. Contributions of the Dissertation . . . . . . . . . . . . . . . . . . . . . . 13
1.5. Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . 14
2. Output Feedback Sliding Mode Control for Sampled-Data Systems 16
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3. Discrete-time Regular Form . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4. One-Step Delayed Disturbance Approximation Approach . . . . . . . . . 24
2.4.1. Output Feedback Control Design . . . . . . . . . . . . . . . . . . 24
2.4.2. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3. Accuracy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5. Two-Step Delayed Disturbance Approximation Approach . . . . . . . . 34
2.5.1. Output Feedback Control Design . . . . . . . . . . . . . . . . . . 34
2.5.2. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.3. Accuracy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6. Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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2.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3. Sliding Mode Control for Singularly Perturbed Linear Continuous-
Time Systems: Composite Control Approaches . . . . . . . . . . . . . . . 47
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1. Dominating Slow Dynamics Approach . . . . . . . . . . . . . . . 57
3.3.2. Dominating Fast Dynamics Approach . . . . . . . . . . . . . . . 62
3.4. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4. Sliding Mode Control for Singularly Perturbed Linear Continuous-
Time Systems: Lyapunov Approaches . . . . . . . . . . . . . . . . . . . . . 80
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1. Dominating Slow Dynamics Approach . . . . . . . . . . . . . . . 83
4.3.2. Dominating Fast Dynamics Approach . . . . . . . . . . . . . . . 85
4.4. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5. Sliding Mode Control for Singularly Perturbed Linear Discrete-Time
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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5.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.1. Dominating Slow Dynamics Approach . . . . . . . . . . . . . . . 102
5.3.2. Dominating Fast Dynamics Approach . . . . . . . . . . . . . . . 105
5.4. Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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List of Figures
2.1. Evolution of the control law for the one-step delayed disturbance approx-
imation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2. O() bounds of the state variables for the one-step delayed disturbance
approximation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3. O(2) accuracy of the sliding motion for the one-step delayed disturbance
approximation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4. Evolution of the control law for the two-step delayed disturbance approx-
imation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5. O(2) bounds of the state variables for the two-step delayed disturbance
approximation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6. O(3) accuracy of the sliding motion for the two-step delayed disturbance
approximation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1. Evolution of the slow state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2. Evolution of the fast state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3. Sliding function evolution for the dominating slow dynamics approach . 69
3.4. Evolution of the composite control law for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5. Evolution of the slow state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6. Evolution of the fast state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7. Evolution of the sliding function for the dominating fast dynamics approach 72
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3.8. Evolution of the composite control law for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9. Evolution of the slow state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.10. Evolution of the fast state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.11. Sliding function evolution for the dominating slow dynamics approach . 76
3.12. Evolution of the composite control law for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.13. Evolution of the slow state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.14. Evolution of the fast state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.15. Evolution of the sliding function for the dominating fast dynamics approach 78
3.16. Evolution of the composite control law for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1. Evolution of the slow state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2. Evolution of the fast state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3. Sliding function evolution for the dominating slow dynamics approach . 91
4.4. Evolution of the sliding mode control law for the dominating slow dy-
namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5. Evolution of the slow state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6. Evolution of the fast state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7. Sliding function evolution for the dominating fast dynamics approach . 93
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4.8. Evolution of the sliding mode control law for the dominating fast dy-
namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.9. Evolution of the slow state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.10. Evolution of the fast state variables for the dominating slow dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.11. Sliding function evolution for the dominating slow dynamics approach . 96
4.12. The evolution of the sliding mode control law for the dominating slow
dynamics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.13. Evolution of the slow state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.14. Evolution of the fast state variables for the dominating fast dynamics
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.15. Sliding function evolution for the dominating fast dynamics approach . 98
4.16. Evolution of the sliding mode control law for the dominating fast dy-
namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1. Evolution of the slow state variables for the dominating slow dynamics
approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2. Evolution of the fast state variables for the dominating slow dynamics
approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3. Sliding function evolution for the dominating slow dynamics approach. . 113
5.4. Evolution of the composite control law for the dominating slow dominat-
ing approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5. Evolution of the slow state variables for the dominating fast dynamics
approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.6. Evolution of the fast state variable for the dominating fast dynamics
approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.7. Sliding function evolution for the dominating fast dynamics approach. . 116
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5.8. Evolution of the composite control law for the dominating fast dynamics
approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
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1
Chapter 1
Introduction
1.1 Sliding Mode Control
Variable structure systems (VSS) are special structure systems that have been exten-
sively studied for decades. The basic philosophy of the variable structure approach is
that the structure of the system varies under certain conditions from one to anothermember of a set of admissible continuous time functions (Utkin, 1977). A VSS can
inherit combined useful properties from the structures. In addition, it can be endowed
with special properties which are not present in any of the structures (Utkin, 1977).
Sliding mode control is a type of variable structure control, where sliding surfaces
or manifolds are designed such that system trajectories exhibit desirable properties as
confined to these manifolds. A system using a sliding mode control strategy can display
a robust performance against parametric uncertainties and exogenous disturbances.This property is of extreme importance in practice where most of plants are heavily
affected by parametric and external disturbances.
Consider a general VSS described by
x(t) = f(x(t), t , u(t)) (1.1)
where x(t) Rn, and u(t) Rm. Each component of control is assumed to act indiscontinuous fashions based on some appropriate conditions,
ui(t) =
u+i (x, t) if si(x) > 0
ui (x, t) if si(x) < 0
, i = 1,...,m (1.2)
where si(x) plays the role of a sliding surface.
Since differential equations (1.1), (1.2) have discontinuous right hand sides, they do
not meet the classical requirements on the existence and uniqueness of solutions. A
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2
formal technique called the equivalent control method was introduced by Utkin (1977)
to analyze the equivalent dynamics of the closed-loop system. In this approach, a
control called the equivalent control is obtained by solving s(t) = 0, namely
ds
dt =s
x .dx
dt =s
x f(x, ueq, t) = 0. (1.3)
The system dynamics on the sliding surface is studied by substituting the equivalent
control ueq in equation (1.1). Note that the equivalent control ueq is not physically real-
izable due to the unknown disturbances. Furthermore, the equivalent system dynamics
is not exactly but very close to the sliding dynamics (Utkin, 1977).
The existence of the sliding mode is described by the following conditions (Utkin,
1978)
Sliding condition (sufficient, local)
lims0+
s < 0, lims0
s > 0 (1.4)
Reaching condition (sufficient, global)
s < sgn(s), (1.5)
where > 0 is a parameter to be designed.
Control magnitude constraint (necessary)
umin ueq umax (1.6)
In sliding mode control, a sliding surface is first constructed to meet existence con-
ditions of the sliding mode. Then, a discontinuous control law is sought to drive the
system state to the sliding surface in a finite time and stay thereafter on that surface.
We now present some fundamental designs of sliding mode control for regular linear
systems. Consider a linear system
x(t) = Ax(t) + Bu(t) + Df(t) (1.7)
where x(t) Rn is the state, u(t) Rm is the control, f(t) Rr is the unknown butbounded exogenous disturbance f M, with m p < n. It is assumed that B is
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3
of full rank, (A, B) is controllable and the following matching condition (Drazenovic,
1969) is satisfied
rank[B|D] = rankD (1.8)
In other words, D can be written as D = BG.
There are several systematic methods for constructing sliding surfaces, such that
Utkin and Youngs method (Utkin and Young, 1979), Lyapunov method (Gutman and
Leitmann, 1976; Gutman, 1979; Su et al., 1996). Utkin and Youngs method and
Lyapunov method to be presented below will be employed in solving our problem in
the following chapters.
Since B has full rank, there exists a nonsingular transformation T such that
T B =
0(nm)m
B0
,
x1(t)
x2(t)
= T x(t),
which bring (1.7) into the normal form
x1(t)
x2(t)
=
A11 A12
A21 A22
x1(t)
x2(t)
+
0
B0
u(t) +
0
B0G
f(t) (1.9)
where x1(t) Rnm, x2(t) Rm. Note that B0 is an mm matrix and it is nonsingular
because B is of full rank.
Regard x2(t) as a control input to the first subsystem of (1.9)
x1(t) = A11x1(t) + A12x2(t) (1.10)
and construct a state feedback gain for (1.10) as
x2(t) = Kx1(t) (1.11)
Hence, the sliding surface in the (x1, x2) coordinate can be chosen as
[K Imm]
x1
x2
= 0 (1.12)
or
s(t) = Cx(t) = [K Imm]T x(t) = 0 (1.13)
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4
in the original coordinates. Once the sliding surface coefficient matrix C is designed,
one can proceed to construct the sliding mode control law. Taking the derivative of
(1.13) with respect to t, we have
s(t) = Cx(t) = CAx(t) + CBu(t) + CDf(t) (1.14)
It is important to emphasize that the matrix CB in (1.14) is an m m nonsingularmatrix equal to B0 since
CB = [K I]T B = [K I]
0
B0
= B0. (1.15)
A sliding mode control law can be designed by using the unit control method or the
signum method (Utkin, 1984). From (1.15), a unit control law can be chosen as (Utkin,1984)
u(t) = (CB )1CAx (CB )1(+ ) ss (1.16)
where
= CDM (1.17)
is a value that helps to tackle disturbances. It can be shown that (1.16) satisfies the
vector form of the reaching condition, that is
sTs = s s + sTCDd(t) < s (1.18)
where is chosen as in (1.17) and is a design parameter for adjusting the reaching
time. One can find the finite reaching time by considering the Lyapunov function
V = sTs (1.19)
Taking its derivative, we have
V < 2s = 2V (1.20)
This yields
dVV
< 2dt. (1.21)
Hence, V(t)
V(0) < t. (1.22)
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5
Let t = Tr be the time to reach the sliding mode (V(Tr) = 0). Thus, the reaching time
is bounded as
Tr 0. (1.24)
A sliding surface is chosen as
s(t) = HB
T
P x(t) = 0. (1.25)
where H is an m m nonsingular matrix. It was proven that the system (1.7) withthe sliding mode on the sliding surface (1.25) is asymptotically stable (Su et al., 1996).
The idea of employing Lyapunovs second method to construct a sliding surface can be
extended to nonlinear systems (Su et al., 1996).
1.2 Singularly Perturbed Systems
Singularly perturbed systems are systems that possess small time constant, or similar
parasitic parameters which usually are neglected due to simplified modeling. When
taking into account those small quantities, the order of the model is increased and the
computation needed for control design can be expensive and even ill-conditioned. How-
ever, if one uses a simplified model to design a control strategy, the desired performance
may not be achieved or the system can be unstable. As a result, singular perturbation
methods have been developed for years to address the stability and robustness of those
systems. For an extensive study, we refer to (Kokotovic et al., 1986; Gajic and Lim,
2001).
Consider a linear singularly perturbed system without control
x(t) = A11x(t) + A12z(t), x(t0) = x0
z(t) = A21x(t) + A22x(t), z(t0) = z0, (1.26)
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where x(t) and z(t) are respectively slow and fast state variables and is a small positive
parameter.
To analyze the system (1.26), one common way is to use the Chang transformation
to transform (1.26) into a block-diagonal system where the slow and fast dynamics are
completely decoupled (Chang, 1972). The Chang transformation is represented by x(t)
z(t)
=
I1 H
L I2 LH
(t)
(t)
= T1
(t)
(t)
(1.27)
and its inverse transformation is given by (t)
(t)
=
I1 HL H
L I2
x(t)
z(t)
= T
x(t)
z(t)
. (1.28)
where L and H matrices satisfy algebraic equations
A21 A22L + LA11 LA21L = 0 (1.29)
and
(A11 A12L)H H(A22 + LA12) + A12 = 0. (1.30)
Matrices L and H can be found using several methods. For example, the Newton
method is presented in (Grodt and Gajic, 1988). The resulting decoupled form is
(t)(t)
= A11 A12L 00 A22 + LA12
(t)(t)
. (1.31)
Now we present a short summary on the design of state feedback control for deter-
ministic linear continuous time singularly perturbed systems. Consider the following
controlled system
x(t) = A11x(t) + A12z(t) + B1u(t), x(t0) = x0
z(t) = A21x(t) + A22z(t) + B2u(t), z(t0) = z0. (1.32)
where x(t) Rn1, z(t) Rn2, and u(t) Rm. The system (1.33) is approximatelydecomposed into an n1 dimensional slow subsystem and an n2 fast subsystem by setting
= 0 in (1.32). The slow subsystem is
xs(t) = Asxs(t) + Bsus(t), xs(t0) = x0
zs(t) = A122 (A21xs(t) + B2us(t), (1.33)
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where
As = A11 A12A122 A21, Bs = B1 A12A122 B2 (1.34)
and the vectors xs(t), zs(t), and us(t) denote the slow parts of x(t), z(t) and u(t). The
fast subsystem is
zf(t) = A22zf(t) + B2uf(t), zf(t0) = z0 zs(t0), (1.35)
where zf(t) = z(t) zs(t), and uf(t) = u(t) us(t) describe the fast parts of thecorresponding variables z(t) and u(t). A composite control law consists of slow and fast
parts as
u(t) = us(t) + uf(t) (1.36)
where us(t) = G0xs(t), and uf(t) = G2zf(t) are independently constructed for the
slow and fast subsystems (1.33) and (1.35). G0 and G2 can be designed by using
classic control theory with an assumption that (As, Bs) and (A22, B2) is controllable.
Nonetheless, a realizable control law must be presented in terms of the actual system
states x(t) and z(t). Replacing xs(t) by x(t) and zf(t) by z(t)zf(t) bring the compositecontrol (1.34) into the realizable feedback form as follows.
u(t) = G0x(t) + G2[z(t) + A122 (A21x(t) + B2G0x(t))] = G1x(t) + G2z(t) (1.37)
where
G1 = (I1 + G2A122 B2)G0 + G2A
122 A21. (1.38)
The discrete-time version of singularly perturbed systems is described in (Litkouhi
and Khalil, 1985). Consider the difference equation
x1[k + 1]
x2[k + 1]
=
(I1 + A11) A12
A21 A22
x1[k]
x2[k]
+
B1
B2
u[k] (1.39)
where x1 Rn1 , x2 Rn2, u Rm, > 0 is a small parameter and det[I2 A22 ] = 0.As in the continuous version, the slow and fast parts of equation (1.39) without control
can be separated by a decoupling transformation (Litkouhi and Khalil, 1985). A control
law which consists of slow and fast components is in the form
u[k] = us[k] + uf[k]
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where uf[k] decays exponentially. To investigate the slow subsystem, we neglect uf[k].
The resulting equations are given by
x1[k + 1] = (I1 + A11)x1[k] + A12x2[k] + B1us[k] (1.40)
x2[k] = A21x1[k] + A22x2[k] + B2us[k]. (1.41)
From (1.40) and (1.41), the slow subsystem is defined by
xs[k + 1] = (I1 + As)xs[k] + Bsus[k] (1.42)
where
xs = x1, (1.43)
As = A11 + A12(I2 A22)1A21, (1.44)
Bs = B1 + A12(I2 A22)1B2. (1.45)
If the pair (As, Bs) is stabilizable in the continuous sense, i.e., every eigenvalue of As
which lies in the closed right-half complex plane is controllable (Litkouhi and Khalil,
1985), then a state feedback control law for us[k] is designed as us[k] = Fsxs[k] where
Fs is chosen such that
Re{(As + BsFs)} < 0. (1.46)
With this choice, the closed-loop slow subsystem system is asymptotically stable. This
shows that the actual design problem for the discrete-time slow subsystem is a contin-
uous one (Litkouhi and Khalil, 1985).
The fast subsystem is defined by assuming that the slow variables are constant
during the fast transient, i.e., x[k + 1] = x[k], and us[k + 1] = us[k]. From (1.41) and
(1.39), the fast subsystem is given by
xf[k + 1] = Afxf[k] + Bfuf[k] (1.47)
where xf = x2 x2, Af = A22, and Bf = B2. If the pair (Af, Bf) is stabilizable in thediscrete-time sense, i.e., every eigenvalue of Af which lies outside or on the unit circle
is controllable (Litkouhi and Khalil, 1985), then a state feedback control law for uf[k]
is given by uf[k] = Ffxf[k] where Ff is chosen such that
|(Af + BfFf)| < 1. (1.48)
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As a result, the closed-loop fast subsystem is asymptotically stable. A composite control
law is taken as the sum of the slow and fast control components
u[k] = Fsxs[k] + Ffxf[k] = Fsxs[k] + Ff(x2[k] x2[k])
= [Fs Ff(I2 Af)1(A21 + BfFs)]x1[k] + Ffx2[k]. (1.49)
It was shown that under the composite control law (1.49), the closed-loop full-order
system is asymptotically stable for sufficiently small . Like in the continuous case, a
stabilizing state feedback control law can be synthesized from slow and fast controllers
that are designed independently.
1.3 Literature on Relevant Works
It is pointed out that for a regular system, a sliding mode control strategy can reject
disturbances and produce a robust performance. In systems with slow and fast modes,
little work has been devoted to the study of sliding mode control (Yue and Xu, 1996;
Su, 1999).
Yue and Xu (1996) studied a singularly perturbed system as follows
x(t) = A11x(t) + A12z(t) + B1u(t) + B1f(t,x,z),
z(t) = A21x(t) + A22z(t) + B2u(t) + B2g(t,x,z), (1.50)
where x(t) Rn1, z(t) Rn2, and u(t) Rm. f(t,x,z), g(t,x,z) : R+Rn1 Rn2 R denote the parameter uncertainties and external disturbances. 0 < < 1 represents
the singular perturbation parameter. Furthermore, the disturbances f(t,x,z), g(t,x,z)
are assumed to satisfy the following inequalities:
|f(t,x,z) 1(x, z) = a0 + a1x + a2z
g(t,x,z 2(x, z) = b0 + b1x + b2z.In addition, they satisfy
f(t,x,z) g(t,x,z) x + z.
In their approach, a designed control law includes two continuous time state feedback
terms and a switching term. The objective of the two continuous-time terms is to
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stabilize the system as no disturbances are taken into account. Specifically, the control
law is in the form
u = Kx + K0 + w (1.51)
where K and K0 are designed such that As + BsK and A22 + B2K0 are stable, and w
is a switching term to be defined. Here, is a new state variable given by
= z + A122 (A21 + B2K)x. (1.52)
To choose w, Yue and Xu (1996) considered a Lyapunov function candidate as follows
V = xTP1x + TP2
where P1, P2 are positive definite solutions to the following Lyapunov equations
(As + BsK)TP1 + P1(As + BsK) = Q1
(A22 + B2K0)TP2 + P2(A22 + B2K0) = Q2.
A control law is chosen as
u = Kx K0 (b0 + b01x + b2)sgn(BT1 P1x + BT2 P2) (1.53)
where b0, b01, and b2 are acquired from the definition of disturbances f(t,x,z) and
g(t,x,z) and matrices A21, A22, K. With this control law, the trajectories x and
ultimately satisfy (Yue and Xu, 1996)
x O(), O().
Yue and Xu (1996) also proved the existence of the sliding motion for the sliding
surface s = BT1 P1x + BT2 P2 provided some condition are satisfied. They employed
Lyapunov functions to construct the sliding function and the sliding mode control law.
Although, their approach deals with disturbances and provides some certain robustcharacteristics, they only guarantee that the trajectories of the system stay in an O()
boundary of the origin. Furthermore, it is somewhat complicated to compute some
parameters for their control law.
Heck (1991) studied a singularly perturbed system in the form of (1.50) without any
disturbances. The full-order system is separated into slow and fast subsystems (1.33),
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(1.35). A sliding-mode controller is constructed for each subsystem. Specifically, a slow
sliding surface can be chosen as
ss(xs) = Ssxs = 0
and a fast sliding surface is taken as
sf(zf) = Sfzf = 0.
Corresponding slow and fast control laws are designed for these sliding surfaces. A
composition of these control law is then implemented for the full-order system. Stability
analysis was carried out by using the equivalent method of Utkin (1977). It was shown
that the reduced-order subsystems approximate the full-order system with accuracy
O(). If reaching conditions are satisfied for the reduced-order models and an additional
condition is met, the reaching conditions are satisfied for the full-order system (Heck,
1991). One draw back of Hecks approach is that the boundedness of the derivative of
the slow sliding mode control must be supposed. Furthermore, parameter uncertainties
and external disturbances were not taken into consideration. As a result, Hecks scheme
is limited in real applications.
Li et al. (1995a) also considered a singularly perturbed system in the form of (1.32).
Similarly to Hecks approach, the full-order system is first decomposed into slow and
fast subsystems. Then, slow and fast sliding mode controllers are designed for the
subsystems individually. The composite control consists of two terms
u = ueq + u
where ueq is the equivalent control for the full-order system and u is the switching
term which is the regulating control moment for the full-order system. A sigmoid
function is exploited to eliminate the chattering phenomenon. Although the switching
surface of the full-order system is decided by the fast switching surface of the reduced-
order system, when reverting back to the original coordinates, the slow sliding control
still exists in the composite one. Like Hecks method, their approach does not address
uncertainties and external disturbances.
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Su (1999) studied the problem of sliding surface design for the system (1.32). Like
(Heck, 1991; Li et al., 1995a), the full-order system is separated into slow and fast
subsystems. Then, stabilizing state feedback controllers are constructed individually
for each subsystem, leading to a composite control law (1.36). The closed-loop system
is transformed into an exact slow and an exact fast subsystems by using the Chang
transformation (Chang, 1972; Kokotovic et al., 1986). The exact subsystems in the
new coordinates are
= Ts
= Tf.
There exist positive definite matrices Ps and Pf such that
PsTs + TTs Ps = Qs, Qs > 0
PfTf + TTf Pf = Qf, Qf > 0.
Then, the sliding surface for the singularly perturbed system can be chosen as
s(x, z) =
B1
B2/
T
JT
Ps 0
0 Pf
J
x
z
= 0.
It was shown that (Su, 1999) if the sliding motion is achieved, the system is asymptot-
ically stable. However, a control strategy has not been provided to realize the sliding
motion.
There have been several approaches to deal with the problem of sliding mode control
for singularly perturbed systems. Many of them (Heck, 1991; Li et al., 1995a; Su, 1999)
do not address parameter uncertainties and external disturbances which are inherent
in many practical plants. Heck (1991); Li et al. (1995a) exploited the decomposition
of the full-order system into slow and fast subsystems in an effort to construct sliding
surfaces and sliding mode controls for each subsystems. Meanwhile, Su (1999) only
took advantage of the structure of the closed-loop system under the composite control
law, by which a sliding surface is designed. Although Yue and Xu (1996) dealt with
parameter uncertainties and external disturbances, their method is not able to reject
disturbances completely. Instead, the trajectories of the system are ensured in an O()
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bound of the origin. One common feature of the above approaches is that Lyapunov
functions are employed to construct sliding surfaces and sliding mode controls.
1.4 Contributions of the Dissertation
The contributions of the dissertation are summarized in the following:
Two methods for design of an output feedback sliding mode control law forsampled-data systems are proposed in Chapter 2. The first method employs the
one-step predictor ahead technique to approximate the external disturbance. The
second method offers better way to approximate the disturbance by using system
information from two previous time instants. The main feature of the proposed
methods is that the control law is high gain, which leads to singular perturbation
behavior of the closed-loop system. The characteristics of the closed-loop systems
are much better than the other works available in the control literature. In addi-
tion, they capture the same level as in the state feedback case. The results in this
topic have been presented at 2009 American Control Conference, 2010 American
Control Conference, and published in IEEE Transactions on Automatic Control.
The development of two composite design methods for a singularly perturbedsystem with external disturbances is used to design a sliding mode controller. A
state feedback control law for either slow or fast modes combines with a sliding
mode control law to constitute a composite control law. The closed-loop system
under the proposed methods is asymptotically stable and robust against exter-
nal disturbances in the sliding mode. This contribution has been accepted for
publication in Dynamics of Continuous, Discrete and Impulsive Systems journal.
Two approaches based on the Lyapunov equations are used to design a slidingmode controller for singularly perturbed systems with external disturbances. The
results obtained are comparable to those of the composite control approaches;
namely, disturbance rejection and asymptotic stability in the sliding mode are
attained. These features show the advantages of the proposed methods when
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compared to other works available in the literature. This contribution is submitted
for journal publication.
The formulation and analysis of a sliding mode control problem for a discrete-
time singularly perturbed systems is investigated. Two composite approaches are
proposed to deal with the stability and robustness of a system with parametric
uncertainties. A state feedback control law is constructed for either slow or fast
modes. The remaining modes are handled by a sliding mode control law. A
composite control law combining the two components provides the system with
asymptotic stability and robustness against modeling uncertainties.
1.5 Organization of the Dissertation
Throughout the dissertation, we deal with different issues of sliding mode control for
singularly perturbed systems: discrete- and continuous-time domains. Each chapter
presents unified techniques or tools to address specific problems.
In Chapter 2, we formulate and develop an output feedback control strategy for
sampled-data systems. Stability and robustness will be analyzed by using singular
perturbation techniques. Furthermore, it will be theoretically shown that the same
characteristics as in the state feedback case are maintained. At the end of the chapter,
the numerical simulation of an aircraft model illustrates the advantages of the proposed
method.
Chapter 3 presents two composite control approaches for the sliding mode control
for singularly perturbed systems with external disturbances. We show how to design a
sliding surface and a corresponding control law to effectively stabilize the system and
reject the external disturbances. Two numerical examples at the end of the chapter
demonstrate the effectiveness of the proposed methods.
The same problem is addressed in Chapter 4 by different approaches. We employ
Lyapunov functions and the Chang transformation to construct a sliding surface and a
sliding mode control law to stabilize the closed-loop system and reject external distur-
bances.
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The techniques of Chapter 3 are applied to singularly perturbed discrete-time sys-
tems with parametric uncertainties in Chapter 5. Two composite sliding mode control
strategies are presented to deal with the stability and robustness of the closed-loop
system. A numerical example of a discrete-time model of a steam power system is
provided to illustrate the efficacy of the methods.
Finally, Chapter 6 presents an overview of the results of the dissertation which is
followed by some future directions.
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Chapter 2
Output Feedback Sliding Mode Control for Sampled-Data
Systems
2.1 Introduction
The problem of output feedback sliding mode control with disturbances has been ex-
tensively studied for years (Zak and Hui, 1993; EL-Khazali and DeCarlo, 1995; Hecket al., 1995; Edwards and Spurgeon, 1995, 1998). It was pointed out that the problem
of choosing the desired poles of the sliding mode dynamics can be approached by using
the classical squared-down techniques (MacFarlane and Karcanias, 1976). In order
to attain a global attraction to the sliding surface, Heck et al. (1995) established nu-
merical methods that adjust the switching gain to compensate for the unknown state
and disturbance variables. Edwards and Spurgeon (1995) proposed a procedure to con-
struct a sliding surface based on the output information by taking advantage of thefact that the invariant zeros of a system appear in the dynamics of the sliding motion.
The remaining eigenvalues of the sliding mode dynamics can be chosen appropriately
in the framework of a static output feedback pole placement problem for a subsystem
(Zak and Hui, 1993; Edwards and Spurgeon, 1995).
In this chapter, we consider the output feedback sliding mode control for sampled-
data linear systems. It is well-known that the exact continual sliding motion cannot be
achieved in the discrete-time case due to the sample/hold effect (Milosavljevic, 1985).Specifically, the system trajectory only travels in a neighborhood of the sliding surface
forming a boundary layer (Su et al., 2000). Several approaches were proposed to address
the problem of discrete-time output feedback sliding mode control (Lai et al., 2007; Xu
and Abidi, 2008a,b). Some of them are devoted to sliding mode control of sampled-data
systems. Xu and Abidi (2008a) employed a disturbance observer and a state observer
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with an integral sliding surface to address the output tracking problem for sampled-
data systems. Under their proposed control approach, the stability of the closed-loop
system is guaranteed and the effect of external disturbances is reduced (Xu and Abidi,
2008b).
The deadbeat control strategy proposed in (Oloomi and Sawan, 1997) is able to
decouple external disturbances with an O() accuracy, where is the sampling pe-
riod. In (Su et al., 2000), an one-step delayed disturbance approximation approach
has been shown to be effective in dealing with disturbances that exhibit certain con-
tinuity characteristics. We shall exploit the continuity attribute of the state variables
and the disturbances by using similar ideas to deal with the similar estimation problem
encountered above.
We will present two dynamic output feedback discrete-time control strategies that
take into account the disturbance compensation as in (Su et al., 2000). In the first
approach, the estimation of the disturbance is based on its previous time instant value,
while the second approach employs two previous time instant values. Both methods
possess high gain output feedback control. It is pointed out that with high gain output
feedback control, the system exhibits the two-time scale behavior (Litkouhi and Khalil,
1985; Gajic and Lim, 2001; Oloomi and Sawan, 1997; Young et al., 1977). By using
singular perturbation analysis, we will study the stability of the closed-loop system and
the accuracy of the sliding mode. Specifically, we will show that the state trajectory will
be remained in an O(2) boundary layer of the sliding surface in the first approach and
an O(3) vicinity of the sliding surface in the second method. While the first approach
shares with (Xu and Abidi, 2008a) some characteristics such as the bound of sliding
motion and the ultimate bound of state variables, the second one presents stronger
results. In addition to a controller for implementation, the method in (Xu and Abidi,
2008a) is performed with two observers for state and disturbance estimation, while we
employ no observers. On the other hand, the method of Xu and Abidi (2008a) applies
to systems of relative degree higher than one, while our methods are limited to systems
with relative degree one.
Throughout the dissertation, {A} denotes the spectrum of matrix A, while Im
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stands for an identity matrix of order m. A vector function f(t, ) Rn is said to beO() over an interval [t1, t2] Kokotovic et al. (1986) if there exists positive constants k
and such that
f(t, )
k
[0, ],
t
[t1, t2]
where . is the Euclidean norm. Moreover, it is said to be O(1) over [t1, t2] if
f(t, ) k, t [t1, t2].
2.2 Problem Formulation
We consider a linear system described by
x0(t) = A0x0(t) + B0u(t) + D0f(t) (2.1)
y(t) = C0x0(t)
where x0(t) Rn is the system state, u(t) Rm is the control, y(t) Rp is thesystem output, f(t) Rr is the unknown but bounded exogenous disturbance, withm p < n. The system matrices A0, B0, C0, D0 are constant of appropriate dimensionswith magnitudes O(1). The following assumptions are made:
1. The system (2.1) has relative degree 1.
2. Matrices B0 and C0 have full rank.
3. (A0, B0, C0) is controllable and observable (EL-Khazali and DeCarlo, 1995).
4. The invariant zeros of system (2.1) are stable.
In addition, D0 satisfies the matching condition (Drazenovic, 1969)
rank([B0
|D0]) = rank(B0) (2.2)
In other words, there exists a matrix K such that
D0 = B0K. (2.3)
The sliding surface under consideration is
s(t) = Hy(t) = HC0x0(t) = 0, (2.4)
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where H is a full rank m p matrix, designed to render stable sliding dynamics. It isshown that the eigenvalues of the sliding mode dynamics include the invariant zeros of
the system (2.1) (Edwards and Spurgeon, 1998). One can place the remaining eigen-
values of the zero dynamics of the sliding surface (2.4) if the Davison-Kimura condition
(Davison and Wang, 1975) is satisfied (Edwards and Spurgeon, 1998). In the case the
Davison-Kimura condition is not satisfied, a dynamic compensator is constructed to
produce a tractable structure for the sliding surface design (Edwards and Spurgeon,
1998). Refer to (EL-Khazali and DeCarlo, 1995; Zak and Hui, 1993; Edwards and
Spurgeon, 1998) for design of H. Note that HC0B0 is nonsingular. Our objective is to
construct a discrete-time sliding mode controller given output sliding surface (2.4).
2.3 Discrete-time Regular Form
In this section, we will use several similarity transformations to facilitate system design
and analysis. Since rank(B0) = m, there exists a coordinate transformation T0 such
that
B = T0B0 =
0
B2
.
where B2 is a nonsingular square matrix of dimension m. The new variables are defined
as
x =
x1
x2
= T0x0.
The similarity transformation T0 brings the original system (2.1) into the regular
form (Utkin and Young, 1979)
x1(t) = A11x1(t) + A12x2(t)
x2(t) = A21x1(t) + A22x2(t) + B2u(t) + D2f(t), (2.5)
where D2 = B2K. The sliding surface (2.4) is now described as
s(t) = HC0x0(t) = HC0T10 x(t) = Cx(t) = C1x1(t) + C2x2(t) = 0, (2.6)
where
HC0T10 = C.
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The zero dynamics of the sliding mode is represented by the eigenvalues of matrix
Ac = A11A12C12 C1. Note that H has been chosen in (2.4) to render a sliding surfacecoefficient matric C such that C2 is invertible and Ac is stable (Utkin and Young, 1979).
Sampling the continuous-time system (2.5) with the sampling period results in the
following discrete-time model
x[k + 1] = x[k] + u[k] + d[k] (2.7)
where
= eA, =
0
eAdB,
d[k] =
0
eABK f((k + 1)
)d. (2.8)
The system matrices, and , of the sampled-data system (2.7) can be reformulated
by taking the Taylor series expansion as
= eA = I+ A +2
2!A2 + = I+ (A + A) = O(1) (2.9)
and
=
0
eAd B = (B + B) = O(), (2.10)
where
A =1
2!A2 +
3!A3 + = O(1) (2.11)
and
B =1
2!AB +
3!A2B + =
B1
B2
= O(1), (2.12)
where the dimensions of B1 and B2 are (n m) m and m m, respectively.
Furthermore, since B =
0
B2
, can be written as
=
2B1
B2
, (2.13)
where
B2 = B2 + B2. (2.14)
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Due to the sampling process, the disturbance in the sampled-data system (2.7) exhibits
unmatched components, as demonstrated by the following lemma, which is an enhanced
version of Abidi et al. (2007).
Lemma 2.1. If the first and second derivatives of the disturbance f(t) in (2.1) are bothwell defined and bounded, then
d[k] =
0
eABK f((k + 1) )d = Kf[k] + 2
Kv [k] + 3d[k],
d[k] d[k 1] = O(2),
d[k] 2d[k 1] + d[k 2] = O(3), (2.15)
where v(t) = df(t)/dt and d[k] is a bounded uncertainty.
Proof. Consider 0 < and express f((k + 1) ) as
f((k + 1) ) = f[k] +(k+1)k
v()d
= f[k] +
(k+1)k
(v[k] +
k
v()d)d
= f[k] + v[k]( ) +(k+1)k
k
v()dd (2.16)
with k < (k + 1) . Substituting (2.16) into the expression of d[k] yields
d[k] =0
eABK f((k + 1) )d=
0
eABK f[k]d +
0
eABK v[k]( )d
+
0
eABK
(k+1)k
k
v()ddd. (2.17)
Following the same steps as in Lemma 1 in Abidi et al. (2007), we obtain0
eABK f[k]d = Kf[k], (2.18)
0 e
A
BK v[k]( )d =
2 Kv [k] +
M v[k]
3
, (2.19)
where M is a constant matrix. Assume the second derivative off(t) is bounded by W,
namely v(t) W. Then, we have
(k+1)k
k
v()dd (k+1)k
k
Wdd
W( )2 < W 2. (2.20)
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Hence,
0
eABK
(k+1)k
k
v()ddd
0
usi(xs), if ssi(xs) 0(3.7)
where ssi = Csixs is the ith linear switching function, u+si and u
si are smooth functions
to be defined later. The equivalent control for the slow subsystem during sliding is
given by Utkin (1984)
ues = (CsB0)1CsA0xs. (3.8)
For the fast subsystem (3.5), a sliding surface is chosen as
sf = Cfzf = 0. (3.9)
The control is chosen in the form:
ufi(zf) =
u+fi(zf), if sfi(zf) > 0
ufi(zf), if sfi(zf) 0(3.10)
where sfi = Cfizf is the ith linear switching function, u+fi and u
fi are smooth functions
to be defined later. The equivalent control for the fast subsystem during sliding is given
by Utkin (1984)
uef = (CfB2)1CfA22zf. (3.11)
The control law for the full-order system is a composite of the slow and fast controls
as given by
u = us(xs) + uf(zf) (3.12)
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where us is defined in (3.7) and uf is defined in (3.10).
The structure of the slow control law (3.7) and the fast control one (3.10) can be
chosen by three methods (Heck, 1991). For a system
x = Ax + Bu (3.13)
the control law is given (Utkin, 1977)
ui = (mi=1
|xm| )sgn(si) (3.14)
where ui is the ith component of the control, si is the ith row vector of s(t). A control
law proposed by DeCarlo et al. (1988) can be used for the slow subsystem
u = (SB)1SAx (SB)1SGN(Sx) (3.15)
where SGN(y) is a vector-valued function with the ith component equal to sgn(yi). The
composite control law can be given by
u = SGN(Kxx + Kzz) (3.16)
where Kx, Kz are to be determined. The main contribution of Heck is that the problem
of sliding mode control for the full order system (3.1) is addressed by two reduced-order
problems for the slow and fast subsystems. The drawback of Hecks method is the
assumption ondzs)dx() to validate the fast model; hence, the generality of the method is
limited. Furthermore, external disturbances were not brought into the picture.
Li et al. (1995a) also considered a singularly perturbed system in the form of (3.1).
Similarly to Hecks approach, the full-order system is first decomposed into slow and
fast subsystems. Then, slow and fast sliding surfaces are designed for the subsystems
separately. The only difference is that Li et al. (1995a) proposed a non switching control
strategy. The control law for the slow subsystem is given by
us = ues + us (3.17)
where ues is defined in (3.8) and
us = (CsB0)1ssig(Csxs) sgn(Csxs) (3.18)
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where stands for the component-wise multiplication, s is a positive scalar, andsig(Csxs) is a vector function whose ith component is given by
sig(Csixs) =1 e|Csixs|1 + e|Csixs|
0. (3.19)
Similarly, the fast sliding mode control law is given by
uf = uef + uf (3.20)
where
uef = (CfB2)1CfA22zf
is equivalent control and
uf = (CfB2)1ssig(Cf) sgn(Cfzf)
The composite control is constructed from slow and fast control laws:
u = us(x) + uf(zf)
where us and uf are defined in (3.17) and (3.20). The control method proposed by Li
et al. (1995a) employs sigmoid functions to reduce the chattering phenomenon. Like
(Heck, 1991), they did not consider the problem of external disturbances beside the
closed-loop stability issue.
Following the same direction as in (Heck, 1991; Li et al., 1995a), Innocenti et al.
(2003) studied a class of singularly perturbed systems:
x(t) = A11x(t) + A12z(t), x(t0) = x0
z(t) = A21x(t) + A22z(t) + B2u(t), z(t0) = z0, (3.21)
where x(t) Rn1, z(t) Rn2, u(t) Rm, is a small positive parameter, A22 isnonsingular, B2 is of full rank, and (A22, B2) is controllable. It is seen that system
(3.21) is less general than system (3.1) when the control input only affects the system
through the dynamics ofz(t). Like in (Heck, 1991), they employed standard techniques
of singular perturbations to decompose system (3.21) into slow and fast subsystems.
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The slow model is given by
xs(t) = A0xs + B0us(xs)
zs = A122 (A21xs + B2us(xs)) (3.22)
where A0 = (A11 A12A122 A21) and B0 = A12A122 B2. The slow sliding surface ischosen as
ss = Csxs = 0.
The slow control law is given by
us(xs) = (CsB0)1(Ksss + Qss(ss) + CsA0xs) (3.23)
where Ks Rrr
, Qs Rrr
are positive definite diagonal matrices and s() : Rr
Rr is a vector function, whose ith component is given by
si(i) =i
|i| + si(3.24)
and si is a small positive quantity. The fast reduced model obtained is the same as in
(Heck, 1991; Li et al., 1995a):
dzfd
= A22zf + B2uf (3.25)
where zf = z zs is the fast component of z and uf = u us is the fast control. If thefast sliding surface is chosen as
sf = Cfzf = 0,
the fast sliding control law is given by
uf(zf) = (CfB2)1(Kfsf + Qff(sf) + CfA22zf). (3.26)
The composite control is synthesized from slow control (3.23) and fast control (3.26)
uc = us(x) + uf(zf).
To study the stability of dynamics of subsystems, Innocenti et al. (2003) employed a
state space decomposition to construct a quadratic Lyapunov function and a procedure
in (Kokotovic et al., 1986). Under their proposed scheme, the closed-loop system is
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proved to be globally stable with sufficiently small . Different from Heck (1991); Li
et al. (1995a), Innocenti et al. (2003) utilized a continuous-time control law instead
of discontinuous one. Hence, it helps avoid chattering phenomena. While external
disturbances are also not considered, the class of systems under consideration is less
general than in (Heck, 1991; Li et al., 1995a).
Unlike the above approaches, where no disturbances are taken into account, Yue
and Xu (1996) studied a singularly perturbed system of the form
x(t) = A11x(t) + A12z(t) + B1u(t) + B1f(t,x,z),
z(t) = A21x(t) + A22z(t) + B2u(t) + B2g(t,x,z), (3.27)
where x(t) Rn1
, z(t) Rn2
, and u(t) Rm
. 0 < 1 represents the singularperturbation parameter. f(x,z ,t), g(x,z ,t) : R+Rn1Rn2 R denote the param-eter uncertainties and external disturbances. Furthermore, the disturbances f(x,z ,t),
g(x,z ,t) are assumed to satisfy the following inequalities:
|f(x,z ,t) 1(x, z) = a0 + a1x + a2z
g(x,z ,t) 2(x, z) = b0 + b1x + b2z.
In addition, they satisfy
f(x,z ,t) g(x,z ,t) x + z.
In their approach, a designed control law includes two continuous time state feedback
terms and a switching term. The objective of the two continuous-time terms is to
stabilize the system as no disturbances are taken into account. Specifically, the control
law is in the form
u = Kx + K0 + w (3.28)
where K and K0 are designed such that As + BsK and A22 + B2K0 are stable, and w
is a switching term to be defined. Here, is a new state variable given by
= z + A122 (A21 + B2K)x. (3.29)
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The new system is given by
x = A011x + (A12 + B1K0) + B1w + B1f
= (A22 + B2K0) + B2w + B2g + O().
To choose w, Yue and Xu considered a Lyapunov function candidate as follows
V = xTP1x + TP2
where P1, P2 are positive definite solutions to the following Lyapunov equations
(As + BsK)TP1 + P1(As + BsK) = Q1
(A22 + B2K0)TP2 + P2(A22 + B2K0) =
Q2.
The control law is chosen as
u = Kx K0 (b0 + b01x + b2)sgn(BT1 P1x + BT2 P2) (3.30)
where b0, b01, and b2 are acquired from the definition of disturbances f(t,x,z), g(t,x,z),
and matrices A21, A22, K. With this control law, the system (3.27) is uniformly prac-
tically stable for (0, ] and the trajectories x and ultimately satisfy (Yue and Xu,
1996)
x O(), O().
Yue and Xu also proved the existence of the sliding motion for the sliding surface s =
BT1 P1x + BT2 P2 provided some condition are satisfied (Yue and Xu, 1996). Although,
their approach deals with disturbances and provides some certain robust characteristics,
it only guarantees the trajectories of the system travel in a O() boundary layer of the
origin. Furthermore, it is complicated to compute some parameters for the control law.
Su (1999) studied the problem of sliding surface design for the system (3.1). Like in
(Heck, 1991; Li et al., 1995a; Innocenti et al., 2003), the full-order system is separated
into slow and fast subsystems. Then, stabilizing state feedback controls are constructed
individually for each subsystem, leading to a composite control law
u = us + uf = K1x + K2z. (3.31)
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Then, the closed-loop system is written as
x
z
=
T11 T12
T21 T22
(3.32)
where
T11 = A11 + B1K1,
T12 = A12 + B1K2,
T21 = A21 + B2K1,
T22 = A22 + B2K2.
The closed-loop system is transformed into an exact slow and an exact fast subsystem
by using the Chang transformation (Chang, 1972; Kokotovic et al., 1986):
=
In1 HL H
L In2
x
z
= J
. (3.33)
The exact subsystems in the new coordinates are
= (T11 T12) = Ts
= (T22 + LT12) = Tf. (3.34)
There exist positive definite matrices Ps and Pf such that
PsTs + TTs Ps = Qs, Qs > 0
PfTf + TTf Pf = Qf, Qf > 0. (3.35)
Then, the sliding surface for the singularly perturbed system can be chosen as
s(x, z) = B1
B2/
T
JT Ps 0
0 Pf
J x
z
= 0. (3.36)
It was shown that (Su, 1999) if the sliding motion is achieved, the system is asymptoti-
cally stable. Like in (Yue and Xu, 1996), only one sliding surface is designed for the full
order system. However, a control strategy has not been provided to realize the sliding
motion.
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In this chapter, we address the problem of sliding mode control for a singularly
perturbed system with the external disturbance. While several papers in the literature
only address the stability of the closed-loop system (Heck, 1991; Li et al., 1995a; Inno-
centi et al., 2003), we consider both closed-loop stability and disturbance rejection. In
our method, a state feedback control law is firstly established to stabilize either slow or
fast dynamics. Then, a sliding mode control law is designed for the remaining dynamics
of the system to ensure stability and disturbance rejection. Putting the two controls
together produces a composite control law that makes the closed-loop system asymp-
totically stable. The advantage of the proposed method over others in the literature is
that external disturbances are completely excluded.
3.2 Problem Formulation
Consider a singularly perturbed system:
x(t)
z(t)
= A
x(t)
z(t)
+ Bu(t) + Df(t), (3.37)
A =
A11 A12
A21 A22
, B =
B1
B2
, D =
D1
D2
,
where x(t) Rn1 and z(t) Rn2 are the slow time and fast time state variables,u(t) Rm is the control input, is a small positive parameter. Matrix A22 is invertible,that is rank(A22) = n2. Matrices A, B, D are constant and of appropriate dimension.
Furthermore, f(t) Rr is an unknown but bounded exogenous disturbance with f(t) h.
Disturbance rejection and parameter variation invariance will be achieved if the
matching condition of Drazenovic (1969) is satisfied:
rank[B|D] = rankB. (3.38)
Due to this invariance condition, there exists an m r matrix G such that
D = BG. (3.39)
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Hence, D1 = B1G and D2 = B2G.
Our objective is to find a sliding mode control law to achieve both system stability
and disturbance rejection.
3.3 Main Results
In this section, we will present two sliding mode control strategies for the singularly
perturbed system (3.37). The two control methods share a similar procedure:
A state feedback control law is designed to maintain either the fast or slow modesasymptotically stable.
A discontinuous sliding mode control law for the remaining modes is establishedto reject disturbances.
The results are synthesized in a composite control law to ensure the stability androbustness of the whole system.
Before proceeding to the main results, we need the following assumption.
Assumption 3.1. (A0, B0) and (A22, B2) are controllable.
A0 = A11 A12A122 A21, B0 = B1 A122 B2.
This assumption allows us to construct state feedback control laws separately for
the slow and fast subsystems provided the original full-order system is controllable.
3.3.1 Dominating Slow Dynamics Approach
In this approach, a linear state feedback control law is designed to place eigenvalues
of the fast subsystem into appropriate positions, and then a sliding mode control law
is used for the slow subsystem to exhibit the desired slow time system performance.
Although the original singularly perturbed system can be decoupled into two time-
scales and into two lower dimensional state vectors, (t) and z(t), the similar type of
decomposition does not hold for the control law and the disturbance. As it can be
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seen in the slow and fast subsystems, the control law u(t) is decomposed only in the
time scale, us(t) and uf(t), but not in its dimension. In other words, both us(t) and
uf(t) are still m-vectors as its composite version u(t) = us(t) + uf(t). The same holds
for the disturbance f(t). Therefore, we allege that when it comes to the subject of
disturbance rejection, we can only have one subsystem that enters the sliding mode
unless an appropriate dimensional decomposition in the control law can be achieved.
With Assumption 3.1, a continuous fast-time state feedback control law is chosen
as
uf(t) = Kfz(t) (3.40)
such that A22 + B2Kf is asymptotically stable. The gain matrix Kf can be chosen
appropriately via the eigenvalue placement technique since (A22, B2) is controllable.Then, the system under the composite control law u(t) = us(t) + uf(t) is defined as
x(t)
z(t)
=
A11 (A12 + B1Kf)
A21 (A22 + B2Kf)
x(t)
z(t)
+
B1
B2
(us(t) + Gf(t)). (3.41)
By the change of variables
(t) = x(t) M()z(t), (3.42)
the system (3.41) is transformed into a lower triangular form (sensor form) (Kokotovic
et al., 1986)
(t)
z(t)
=
As 0
A21 A22 + B2Kf + A21M
(t)
z(t)
+
Bs
B2
(us(t) + Gf(t))
(3.43)
where
As = A11 M A21, (3.44)
Bs = B1 MB2 (3.45)and M is the solution to the following algebraic equation (Kokotovic et al., 1986)
A12 + B1Kf M(A22 + B2Kf) + A11M MA21M = 0. (3.46)
Matrix M can be found either using the fixed-point iterations or the Newton method
(Grodt and Gajic, 1988; Gajic and Lim, 2001).
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The system formulation (3.43) is related with its original form (3.37) via state
feedback (3.40) and the similarity transformation. Therefore, the controllability of the
slow subsystem pair (As, Bs) is intact (Chen, 1998). The design objective of the slow-
time control law us(t) is to stabilize the slow subsystem and simultaneously reject the
disturbance f(t) by applying the sliding mode control technique. We choose a sliding
surface for the dominating slow dynamics using the method of (Utkin and Young, 1979)
as
ss(t) = Cs(t) (3.47)
If m < n1, there exists a transformation Ts Rn1n1 for the slow subsystem of (3.43)such that (Utkin and Young, 1979)
TsBs =
0
Bs0
(3.48)
Under this transformation, the slow subsystem of (3.43) becomes
1(t)
2(t)
=
As11 As12
As21 As22
1(t)
2(t)
+
0
Bs0
(us(t) + Gf(t)) (3.49)
where
1(t)2(t)
= Ts(t).
The variable 2(t) should be regarded as a control input to the dynamic equation
of 1(t). According to (Utkin and Young, 1979), the controllability of (As, Bs) implies
the controllability of (As11, As12). As a result, we can find a gain matrix K1 such that
As11 As12K1 is stable. Then, a sliding surface can be chosen as
ss(t) = K11(t) + 2(t) = 0 (3.50)
Representing the sliding surface in (t) coordinates, we obtain
ss(t) = [K1 In1]Ts(t) = Cs(t) = 0 (3.51)
where Cs = [K1 In1]Ts.
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We are now in a position to design a sliding control law for the sliding surface (3.51)
as Cs is chosen. Taking the derivative of the sliding surface (3.51), we have
ss(t) = CsAs(t) + CsBsus(t) + CsBsGf(t). (3.52)
According to (Utkin and Young, 1979), CsBs is nonsingular ifBs has full rank. Assume
that Bs has full rank. Based on the control method of (Utkin, 1978), we choose a sliding
mode control law us(t) as
us(t) = (CsBs)1(CsAs(t) S1ss(t) + (1 + 1) ss(t)ss(t)) (3.53)
where 1 = CsBsGh, 1 is a positive parameter and matrix S1 is asymptoticallystable. The reaching condition is satisfied since
sTs (t) ss(t) = 1
2sTs (t)P1ss(t) 1ss(t) 1ss(t)
+ sTs (t)CsBsGf(t) < 1ss(t) (3.54)
where
P1 = ST1 S1 > 0. (3.55)
This means that us(t) is able to drive the slow variable (t) to reach the sliding surface
ss
(t) in a finite time and reject the disturbance f(t). In the following, we will estimate
the interval of the reaching time.
Choose a Lyapunov function
V(t) = sTs (t)ss(t). (3.56)
We have
max{P1}V(t) 2(1 + 21)
V(t) V(t) min{P1}V(t) 21
V(t). (3.57)
Let 1 be the time needed to reach the sliding mode (V(1) = 0). Taking the derivative
of (3.57), we have
2
max{P1} ln(max{P1}
V(0) + 21 + 41
21 + 41) 1
2
min{P1} ln(min{P1}
V(0) + 21
21). (3.58)
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In other words, the reaching time of the sliding mode lies in the interval
2
max{P1} ln(max{P1}
sTs (0)ss(0) + 21 + 4121 + 41
) 1
2min{
P1}
ln(min{P1}
sTs (0)ss(0) + 212
1
). (3.59)
Remark 3.1. The sliding mode control law (3.53) offers a flexibility in adjusting the
reaching time. Inequalities (3.59) show that choosing appropriate candidates for S1
(P1), 1, and 1 affects the reaching time. Since 1 and 1 constitute the magnitude of
the control effort in sliding mode, their large values are undesired. Hence, we only need
to pick up a suitable value of S1 (P1) to obtain a fast reaching time.
From (3.40) and (3.53), the composite control is given by
u(t) = Kfz(t) (CsBs)1(CsAs(t) S1ss(t) (1 + 1)ss(t)
s(t)). (3.60)
In terms of the original state variables, the control law is rewritten as
u(t) =Kfz(t) (CsBs)1(CsAs(x(t) M z(t)) S1Cs(x(t)
Mz(t)) (1 + 1) Cs(x(t) M z(t))Cs(x(t) M z(t))). (3.61)
When the sliding mode is achieved, one can use the equivalent control method
(Utkin, 1977) to study the dynamics of the closed-loop system. The stability of the
closed-loop system is guaranteed by the following theorem.
Theorem 3.1. There exists > 0 such that, in the sliding mode, the closed-loop
system is asymptotically stable for (0, ] and invariant to the external disturbancef(t).
Proof. To study the dynamics of the closed-loop system under the control law (3.60)
or (3.61), we employ the equivalent control method of (Utkin, 1977). The equivalent
control of the sliding motion is defined by solving ss(t) = 0. This yields
ueqs (t) = (CsBs)1(CsAs(t) + CsBsGf(t)). (3.62)
Substituting the equivalent control into (3.43) results in the following equivalent dy-
namics (t)
z(t)
= 1
(t)
z(t)
(3.63)
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where
1 =
As Bs(CsBs)1CsAs 0
A21 B2(CsBs)1CsAs A22 + B2Kf + A21M
.
The dynamics of the system (3.63) is defined by the eigenvalues of AsBs(CsBs)1CsAsand A22+B2Kf+A21M. Matrix AsBs(CsBs)1CsAs contains m zeros correspondingto the sliding motion and n1 m stable eigenvalues. Since matrix A22 + B2Kf isasymptotically stable, there exists a small 0 such that for all [0, ] theeigenvalues of A22 + B2Kf + A21M have negative real parts. As a result, the closed-
loop system is asymptotically stable according to (Utkin, 1977).
Remark 3.2. One can only study the stability of the slow dynamics by applying the
equivalent control to the dynamics equation (3.49). However, according to (3.43), the
whole transformed system is still affected by us. Hence, when proving stability of the
closed-loop system, we still need to consider the influence of us on the whole system.
3.3.2 Dominating Fast Dynamics Approach
This approach presents a composite control law that consists of slow state feedback
control and fast sliding mode control. According to Assumption 3.1, (A0, B0) is con-
trollable. Thus, there exists a gain matrix Ks such that state feedback control
us(t) = Ksx(t) (3.64)
renders matrix A0 + B0Ks asymptotically stable. The system under the control law
u(t) = us(t) + uf(t) is described as
x(t)z(t)
=A11 + B1Ks A12
A21 + B2Ks A22
x(t)z(t)
+B1
B2
(uf(t) + Gf(t)). (3.65)
Introducing the change of variables
(t) = z(t) + L()x(t) (3.66)
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the system (3.65) is brought into the actuator form (Kokotovic et al., 1986)
x(t)
(t)
=
A11 + B1Ks A12L A12
0 Af
x(t)
(t)
+
B1
Bf
(uf(t) + Gf(t))
(3.67)
where
Af = A22 + LA12 (3.68)
Bf = B2 + LB1 (3.69)
and L is the solution to the following algebraic equation (Kokotovic et al., 1986)
A21 + B2Ks A22L + L(A11 + B1Ks) LA12L = 0. (3.70)
This equation can be solved using either the fixed-point iterations or the Newton method
(Grodt and Gajic, 1988). Since L = A122 (A21 + B2Ks) + O() (Kokotovic et al., 1986),
we have
A11 + B1Ks A12L = A0 + B0Ks + O() (3.71)
Because A0 + B0K is asymptotically stable, there exists a small > 0 such that for
all [0, ], A11 + B1Ks A12L is asymptotically stable.
We are now in a position to construct a sliding mode control law for the fast sub-system in (3.67). Employing the same technique as in the dominating slow dynamics
approach, we choose a sliding surface via the method of (Utkin and Young, 1979) as:
sf(t) = Cf(t) (3.72)
If m < n2, there exists a transformation Tf Rn2n2 for the fast subsystem of (3.67)such that Utkin and Young (1979)
TfBf = 0
Bf0
. (3.73)
Under this transformation, the fast subsystem of (3.67) becomes
1(t)
2(t)
=
Af11 Af12
Af21 Af22
1(t)
2(t)
+
0
Bf0
(uf(t) + Gf(t)). (3.74)
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Consider 2(t) as a control input to the dynamic equation of 1(t). Since (Af21, Af22)
is controllable, we can find a gain matrix K2 such that As21 As22K2 is asymptoticallystable. Then, a sliding surface can be chosen as
sf(t) = K21(t) + 2(t) = 0. (3.75)
Representing this sliding surface in the previous coordinates, we obtain
sf(t) = [K2 In2]Tf(t) = Cf(t) = 0 (3.76)
where Cf = [K2 In2]Tf.
Taking the derivative of the sliding surface (3.76) with respect to t, we have
sf(t) = CfAf(t) + CfBfuf(t) + CfBfGf(t). (3.77)
With the assumption that the disturbance f(t) is bounded, and Bf has full rank, a
control law uf(t) can be chosen as
uf(t) = (CfBf)1(CfAf(t) S2sf(t) + (2 + 2) sf(t)sf(t)) (3.78)
where 2 = CfBfGh, 2 is a positive parameter, and matrix S2 is asymptoticallystable. The reaching condition is satisfied since
sTf(t) sf(t) = 12 sTf(t)P2sf(t) 2sf(t) 2sf(t) + sTfCfBfGf(t) < 2sf(t)
(3.79)
where
P2 = ST2 S2 > 0. (3.80)
Similarly to (3.59), the reaching time of the sliding mode satisfies
2
max{P2}ln(
max{P2}
sT2 (0)s2(0) + 22 + 42
22 + 42)
2
2min{P2} ln(
min{P2}
sT2 (0)s2(0) + 22
22). (3.81)
Like in the first approach, the reaching time of the sliding mode can be monitored
by choosing a suitable value of P2 (S2) without affecting the magnitude of the control
effort during sliding.
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The composite control law is described in terms of the slow state feedback law and
the fast sliding mode control law as follows
u(t) =Ksx(t) (CfBf)1(CfAf(t) S2sf(t) (2 + 2) sf(t)
sf(t)
). (3.82)
In the original coordinates, the control law (3.82) is
u(t) =Ksx(t) (CfBf)1(CfAf(z(t) + Lx(t)) S2Cf(z(t) + Lx(t))
(2 + 2)Cf(z(t) + Lx(t))
Cf(z(t) + Lx(t))). (3.83)
The stability of the closed-loop system under the control law (3.83) is proved in the
following theorem.
Theorem 3.2. Assume(A, B) is controllable. Then there exists > 0 such that in the
sliding mode, the closed-loop system is asymptotically stable for (0, ] and invariantto the external disturbance f(t).
Proof. Like the dominating slow dynamics approach, we employ the equivalent control
method to study the dynamics of the closed-loop system. The equivalent control of the
sliding motion of (3.75) is defined by solving sf(t) = 0 as follows
ueqf (t) = (CfBf)1CfAf(t) Gf(t). (3.84)
Hence, the equivalent dynamics of the closed-loop system under the equivalent con-
trol law (3.84) is given by x(t)
(t)
= 2
x(t)
(t)
(3.85)
where
2 = A11 + B1Ks
A12L A12
B1(CfBf)1CfAf
0 Af Bf(CfBf)1CfAf
.
The dynamics of the system (3.85) is specified by the eigenvalues of A11 + B1KsA12Land 1 (Af Bf(CfBf)1CfAf). The eigenvalues of matrix Af Bf(CfBf)1CfAfinclude m zeros corresponding to the sliding motion and n1 m asymptotically stableeigenvalues. In addition, A11 + B1Ks A12L is asymptotically stable. As a result, the
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dynamics of the system (3.85) is represented by n1 + n2 m stable eigenvalues and mzeros. According to (Utkin, 1977), the closed-loop system is asymptotically stable. This
implies that